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Created by Knut M. Synstadfrom the Noun Project

Local Martingale

Definition (Local Martingale)

Let X=(Xt)t≥0X=(X_{t})_{t\ge 0} be a process on (Ω,F,P)(\Omega,\mathcal{F},P) and let (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0} be a filtration. Let 1≤p<∞1\le p<\infty. (Xt)t≥0(X_{t})_{t\ge 0} is called a local LpL^{p}-martingale if

  1. X0X_{0} is F0\mathcal{F}_{0}-measurable
  2. ∃\exists an increasing sequence (τk)k∈N(\tau_{k})_{k\in\mathbb{N}} of (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}-stopping times with τk↑∞\tau_{k}\uparrow \infty as k→∞k\to\infty s.t. ∀k∈N\forall k\in\mathbb{N}: (Xtτk−X0)t≥0(X_{t}^{\tau_{k}}-X_{0})_{t\ge 0}is a LpL^{p} (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}-martingale. (τk)k∈N(\tau_{k})_{k\in\mathbb{N}} is called a localizing sequence for the local martingale XX.

Remark

X is a Lp-martingale  ⟹  X is a local Lp-martingaleX\text{ is a } L^{p}\text{-martingale}\implies X\text{ is a local }L^{p}\text{-martingale}where τk=+∞\tau_{k}=+\infty ∀k∈N\forall k\in\mathbb{N}.

Proposition (Local Martingale + u.i. = Martingale)

Let (Xt)t≥0(X_{t})_{t\ge 0} be a LpL^{p} (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}-local martingale (1≤p<∞1\le p<\infty) with a localizing sequence (τk)k∈N(\tau_{k})_{k\in\mathbb{N}}. Assume (∣Xtτk∣p)k∈N(|X_{t}^{\tau_{k}}|^{p})_{k\in\mathbb{N}} is uniformly integrable ∀t∈R+\forall t\in\mathbb{R}^{+}. Then (Xt)t≥0(X_{t})_{t\ge 0} is a (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}-martingale i.e. (Xt)t≥0 local martingale and (∣Xtτk∣p)k∈N u.i.   ⟹  (Xt)t≥0 martingale(X_{t})_{t\ge 0}\text{ local martingale and }(|X_{t}^{\tau_{k}}|^{p})_{k\in\mathbb{N}}\text{ u.i. }\implies(X_{t})_{t\ge 0}\text{ martingale}

Linked from

Cross Variation

Itô Stochastic Integral is a Local Martingale

Itô's Formula

Multidimensional Itô Formula

Mutual Variation of Itô Stochastic Integrals

Mutual Variation

Quadratic Variation

Semimartingale

Local Martingale